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\newcommand{\myCourseNumber}{Math 20E}
\newcommand{\myAssignmentDueDate}{Tuesday, December 5, 2023}
\newcommand{\myAssignmentTitle}{Homework Assignment 7}
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\begin{questions}
    
\question Let \ $\mathbf{r}(x,y,z) = (x, \ y, \ z)$ \ and \ $r(x,y,z) = \sqrt{x^2 + y^2 + z^2} = \| \mathbf{r} \|$. \  Verify the following identities.
\begin{parts}
\part $\displaystyle \nabla \left(\frac{1}{r}\right) = -\frac{\mathbf{r}}{r^3}$.
\part $\displaystyle \boldsymbol{\nabla} \boldsymbol{\cdot} \left(\frac{\mathbf{r}}{r^3}\right) = 0$.
\part $\displaystyle \boldsymbol{\nabla} \boldsymbol{\times} \mathbf{r} = \boldsymbol{0}$.
\end{parts}

\question Let $C$ be the closed, piecewise smooth curve formed by traveling in straight lines between the points $(0,0,0)$, $(2,1,5)$, $(1,1,3)$, and back to $(0,0,0)$ in that order.  Use Stokes' theorem to evaluate the line integral
\begin{displaymath} \int_C (x y z) \, dx + (x y) \, dy + (x) \, dz . \end{displaymath}

\question Evaluate the surface integral \ $\displaystyle \iint_S \left( \boldsymbol{\nabla} \boldsymbol{\times} \mathbf{F} \right) \boldsymbol{\cdot} d\mathbf{S}$, \ where $S$ is the portion the surface of a sphere defined by \ $x^2 + y^2 + z^2 = 1$ \ and \ $x + y + z \geq 1$, \ and where \ $\mathbf{F} = \mathbf{r} \boldsymbol{\times} (\mathbf{i} + \mathbf{j} + \mathbf{k})$, \ with \ $\mathbf{r} = x \, \mathbf{i} + y \, \mathbf{j} + z \, \mathbf{k}$.

\question Let \ $\mathbf{F} = x^2 \, \mathbf{i} + (2 x y + x) \, \mathbf{j} + z \, \mathbf{k}$. \ Let $C$ be the circle \ $x^2 + y ^2 = 1$ \ and \ $S$ the disk \ $x^2 + y^2 \leq 1$ \ within the plane \ $z = 0$.
\begin{parts}
\part Determine the flux of $\mathbf{F}$ out of $S$.
\part Determine the circulation of $\mathbf{F}$ around $C$.
\part Find the flux of $\boldsymbol{\nabla} \boldsymbol{\times} \mathbf{F}$.  Verify Stokes' theorem directly in this case.
\end{parts}

%% Rogawski 17.2.20
\question Let \ $\mathbf{F} = (0, \, -z, \, 1)$. \ Let $S$ be the spherical cap \ $x^2 + y^2 + z^2 = 1$, \ where \ $z \geq \frac{1}{2}$.
\begin{parts}
\part Evaluate \ $\displaystyle \iint_S \mathbf{F} \boldsymbol{\cdot} d\mathbf{S}$ \ directly as a surface integral.
\part Verify that \ $\mathbf{F} = \boldsymbol{\nabla} \boldsymbol{\times} \mathbf{A}$, \ where \ $\mathbf{A} = (0, \ x, \ xz)$.
\part Evaluate \ $\displaystyle \iint_S \mathbf{F} \boldsymbol{\cdot} d\mathbf{S}$ \ using Stokes' theorem.
\end{parts}

%% Rogawski 17.2.27
\question Let \ $\mathbf{F} = (y^2, \, x^2, \, z^2)$. \ Verify that \ $\displaystyle \int_{\mathcal{C}_1} \mathbf{F} \boldsymbol{\cdot} d\mathbf{r} = \int_{\mathcal{C}_2} \mathbf{F} \boldsymbol{\cdot} d\mathbf{r}$, \ for any two simple closed curves $\mathcal{C}_1, \mathcal{C}_2$ going around a cylinder whose central axis is the $z$-axis; that is, any cylinder whose equation is of the form \ $x^2 + y^2 = R^2$.

\question Use the divergence theorem to calculate the flux of \ $\mathbf{F} = (x-y) \, \mathbf{i} + (y-z) \, \mathbf{j} + (z-x) \, \mathbf{k}$ \ out of the unit sphere \ $x^2 + y^2 + z^2 = 1$.

\question Let $S$ be the boundary surface of a solid region $W$.  Show that
\begin{displaymath} \iint_S \mathbf{r} \boldsymbol{\cdot} \mathbf{n} \, dS = 3 \ \text{volume}(W) . \end{displaymath}
Explain this result geometrically.

\question Let $W$ be the pyramid with top vertex $(0,0,1)$, and base vertices at $(0,0,0)$, $(1,0,0)$, $(0,1,0)$, and $(1,1,0)$. \ Let $S$ be the closed boundary surface of $W$, oriented outward from $W$. \ Let \ $\mathbf{F}(x,y,z) = (x^2 y, \ 3 y^2 z, \ 9 z^2 x)$. \  Use Gauss' theorem to compute \ $\displaystyle \iint_S \mathbf{F} \boldsymbol{\cdot} \, d\mathbf{S}$.

%% Rogawski 17.3.15
\question Let \ $\mathbf{F}(x,y,z) = (x + y, \ z, \ z - x)$ \ and let $\mathcal{S}$ be the boundary surface of the solid region between the paraboloid \ $z = 9 - x^2 - y^2$ \ and the $xy$-plane.  Evaluate \ $\displaystyle \iint_S \mathbf{F} \boldsymbol{\cdot} d\mathbf{S}$.

\end{questions}

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